Non-uniqueness of certain Hahn—Banach extensions

Beckenstein E.
Владикавказский математический журнал. . Том 6. 2004 г.. Выпуск 1.

Аннотация:
Let \(f\) be a continuous linear functional defined on a subspace \(M\) of a normed space \(X\). If \(X\) is real or complex, there are results that characterize uniqueness of continuous extensions \(F\) of \(f\) to \(X\) for every subspace \(M\) and those that apply just to \(M\). If \(X\) is defined over a non-Archimedean valued field \(K\) and the norm also satisfies the strong triangle inequality, the Hahn—Banach theorem holds for all subspaces \(M\) of \(X\) if and only if \(K\) is spherically complete and it is well-known that Hahn—Banach extensions are never unique in this context. We give a different proof of non-uniqueness here that is interesting for its own sake and may point a direction in which further investigation would be fruitful.

Язык статьи: Английский Загрузить полный текст